Linear response for skew-product maps with contracting fibres

TL;DR

The paper develops a sectional transfer-operator framework for skew-product maps with contracting fibres, enabling invariant measures to be characterized via base dynamics and sample-fibre measures. It proves existence, uniqueness, and differentiability of these invariant objects and derives linear response with respect to system parameters, even in nonuniformly hyperbolic settings. The approach applies to Bernoulli convolution measures and solenoidal attractors with intermittency, including cases where the base map lacks a uniform spectral gap or requires inducing schemes. By combining contraction in fibre space, inducing techniques, and unfolding operators, the authors obtain global, parameter-differentiable linear response results with broad applicability in noninvertible and intermittent dynamical systems.

Abstract

We study linear response for families of skew-product dynamical systems with contracting fibres. Our approach is based on a sectional transfer operator acting on families of probability measures along the fibres. The operator allows to describe invariant measures of the skew-product in terms of sample measures over the base dynamics, regardless of invertibility or non-invertibility of the base map. Under general assumptions, we establish existence and uniqueness of invariant sample measures and prove their differentiability, with respect to system parameters, in suitable topologies. As an application we obtain linear response for Bernoulli convolutions, which are of prime importance in the study of number theoretic problems and fractals. Another application of our results yields linear response for physical measures of solenoidal attractors with intermittency, an example of a hyperbolic system which cannot be handled by traditional transfer operator techniques.

Theorems & Definitions (75)

• Theorem 1
• Theorem 2
• Theorem 3
• Theorem 4
• Remark 1.1
• Theorem 5
• Corollary 6
• Remark 1.2
• Theorem 7
• Theorem 8